Cross Product

The cross product is an operation that takes two vectors in $R^{3}$

and returns another vector in $R^{3}$

, written $a \times b$ . Unlike the dot product, the result is a vector, not a scalar. The cross product is only defined in three dimensions (and in a generalized sense in seven dimensions; here we restrict to $R^{3}$ ).

Geometric meaning

Direction: $a \times b$ is perpendicular to both $a$ and $b$ , following the right-hand rule: if you point your fingers along $a$ and curl them toward $b$ , your thumb points in the direction of $a \times b$ .
Magnitude: $∥ a \times b ∥ = ∥ a ∥ ∥ b ∥ sin θ$ , where $θ$ is the angle between $a$ and $b$ . So the length equals the area of the parallelogram spanned by $a$ and $b$ .

Algebraic definition

For vectors

a = a_{1} a_{2} a_{3}, b = b_{1} b_{2} b_{3},

the cross product is

a \times b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1} .

This can be remembered using the determinant of a formal $3 \times 3$ matrix:

a \times b = e_{1} a_{1} b_{1} e_{2} a_{2} b_{2} e_{3} a_{3} b_{3}

e_{1} (a_{2} b_{3} - a_{3} b_{2}) - e_{2} (a_{1} b_{3} - a_{3} b_{1}) * e_{3} (a_{1} b_{2} - a_{2} b_{1}),

where $e_{1}, e_{2}, e_{3}$ are the standard unit vectors in $R^{3}$ .

The “Cross-Out” Method (Fastest)

The shorthand calculation for this is:

Stack them: Write the components of the first vector over the second vector twice.
Cross out the first and last columns.
Multiply in an ‘X’ pattern (top-left bot-right minus top-right bot-left) for each remaining pair:

Rules of calculation (with examples in LaTeX)

Let $a, b, c \in R^{3}$ and $λ \in R$ .

1. Anticommutativity

Swapping the order flips the sign:

a \times b = - (b \times a) .

Example:

100 \times 010 = 001, 010 \times 100 = 00 - 1 .

2. Distributivity over addition

a \times (b + c) = a \times b + a \times c, (a + b) \times c = a \times c + b \times c .

Example (second component of $a \times (b + c)$ ):

a = 120, b = 011, c = 101 \Rightarrow b + c = 112 .

a \times b = 2 - 1 1, a \times c = 2 - 1 - 2 \Rightarrow a \times b + a \times c = 4 - 2 - 1 .

a \times (b + c) = 2 \cdot 2 - 0 \cdot 1 0 \cdot 1 - 1 \cdot 2 1 \cdot 1 - 2 \cdot 1 = 4 - 2 - 1 .

3. Scalar multiplication (homogeneity)

A scalar can be factored out of either slot:

(λ a) \times b = a \times (λ b) = λ (a \times b) .

Example: with $a = 100$ , $b = 010$ , $λ = 3$ ,

(3 a) \times b = 300 \times 010 = 003 = 3001 = 3 (a \times b) .

4. Cross product with the zero vector

a \times 0 = 0 \times a = 0 .

5. Parallel vectors

$a$ and $b$ are parallel (or one is zero) if and only if

a \times b = 0 .

Example: $a = 246$ , $b = 123 = \frac{1}{2} a$ , so

a \times b = 4 \cdot 3 - 6 \cdot 2 6 \cdot 1 - 2 \cdot 3 2 \cdot 2 - 4 \cdot 1 = 000 .

6. Self-cross product

a \times a = 0 .

(Special case of the parallel-vectors rule.)

7. Jacobi identity

a \times (b \times c) - b \times (c \times a) - c \times (a \times b) = 0 .

8. Relation to dot product (vector triple product expansion)

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c .

Example: $a = e_{1}$ , $b = e_{2}$ , $c = e_{3}$ :

a \cdot c = 0, a \cdot b = 0 \Rightarrow a \times (b \times c) = 0 \cdot b - 0 \cdot c = 0 .

b \times c = e_{1} \Rightarrow e_{1} \times e_{1} = 0 .

9. Magnitude and angle

∥ a \times b ∥^{2} = ∥ a ∥^{2} ∥ b ∥^{2} - (a \cdot b)^{2} .

Equivalently, $∥ a \times b ∥ = ∥ a ∥ ∥ b ∥ sin θ$ .

10. Relation to the dot product (scalar triple product)

a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b) .

This value is the (signed) volume of the parallelepiped spanned by $a, b, c$ . Example:

a = 100, b = 010, c = 001 \Rightarrow b \times c = 100, a \cdot (b \times c) = 1.

Worked example

Compute $u \times v$ for

u = 2 - 1 3, v = 14 - 2 .

u \times v = (- 1) (- 2) - (3) (4) (3) (1) - (2) (- 2) (2) (4) - (- 1) (1) = 2 - 12 3 + 4 8 + 1 = - 10 79 .

Check: $u \cdot (u \times v) = 2 (- 10) + (- 1) (7) + 3 (9) = - 20 - 7 + 27 = 0$ , and $v \cdot (u \times v) = 1 (- 10) + 4 (7) + (- 2) (9) = - 10 + 28 - 18 = 0$ , so the result is perpendicular to both $u$ and $v$ .

Ling's Notes

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